Depolarization Ratio of the ν1 Raman Band of Pure CH4 and Perturbed by N2 and CO2

In this work, the effect of nitrogen and carbon dioxide on the depolarization ratio of the ν1 band of methane in the pressure range of 0.1–5 MPa is studied. A high-sensitivity single-pass Raman spectrometer was used to obtain accurate results. Moreover, we took into account the overlap of the ν1 band by the ν3 and ν2 + ν4 bands using the simulation of their spectra. The depolarization ratio of the ν1 band in pure methane is within 0–0.001, and the effect of nitrogen and carbon dioxide on this parameter is negligible in the indicated pressure range. The obtained results are useful for correct simulation of the Raman spectrum of methane at different pressures, which is necessary to improve the accuracy of gas analysis methods using Raman spectroscopy.


Introduction
The optical methods based on Raman spectroscopy for the analysis of multicomponent gaseous media have been rapidly developing in the last decades. These methods can simultaneously detect all molecular vibrational bands using one laser with a fixed wavelength. The recent appearance of high-sensitivity photodetectors and powerful small-size lasers provides an opportunity to amplify the useful signal and decrease the limit of detection (LOD) of the method. Other amplification methods include the multi-pass optical cells [1][2][3][4] or hollow-core fiber [5][6][7][8]. However, compression of the analyzed medium to a higher pressure is the most effective and easy to implement signal amplification approach [9,10]. Neglecting the compressibility factor, compression of the sample at ambient pressure to a pressure of 5 MPa leads to a 50-fold amplification [11]. The LOD below 1 ppm can be achieved using this approach. Such sensitivity opens up the possibility to analyze the composition of atmospheric and exhaled air using Raman spectroscopy [5, 10,[12][13][14][15][16]. This method is very promising due to its high measurement speed and the ability to determine a lot of compounds.
Methane (CH 4 ) is an important greenhouse gas contained in atmospheric air. The average annual concentration of CH 4 is continuously increasing due to the influence of natural and anthropogenic factors. Therefore, monitoring of atmospheric CH 4 is necessary to detect leaks of greenhouse gases, as well as to improve climate prediction models. The measurement precision of concentration should be less than 100 ppb since CH 4 content in atmospheric air is about 2 ppm [17]. Moreover, CH 4 is included in the list of biomarkers of diseases [18,19]. The CH 4 content in the exhaled air can reach 10 ppm [12]. Hence, high accuracy of CH 4 measurement in a sample is necessary both for the accurate diagnosis and for the investigation of correlations. On the other hand, the region of stretching C-H vibrations (2900-3000 cm −1 ) is important in the air composition analysis since the most intense vibrational Raman bands of all volatile organic compounds (VOCs) are located there [20,21]. The CH 4 content in the air is high compared to other VOCs. Therefore, the most intense fundamental vibrational ν 1 band of CH 4 (≈2917 cm −1 ) makes a significant contribution to the spectrum of these gaseous media in the  Polarized and depolarized spectra of pure CH4, as well as mixtures of CH4/N2 and CH4/CO2 in molar ratios of 50/50, at pressures of 0.1, 0.5, 1, 2, 3, 4, and 5 MPa were recorded using this system. The signal-to-noise ratio in the polarized spectra of pure CH4 was 1500 (at 0.1 MPa) and 11,000 (at 5 MPa), where the peak intensity of the ν1 band (≈2917 cm −1 ) was the signal magnitude. The pressure measurement error was less than 1 kPa. The gas cell was thermally stabilized at 298 ± 1 K. Samples of CH4, N2, and CO2 with a purity of greater than 99.99% were used to prepare the studied mixtures in a separate mixing chamber connected to a gas cell. Pure gases were mixed in a specified ratio of partial pressures to obtain the required molar ratio. These partial pressures were calculated from the equation of state for gases taking into account the compressibility. Compressibility factors were taken from the NIST Chemistry WebBook [46]. The molar ratio measurement error in the mixture preparation procedure is estimated within 2-3%.  Polarized and depolarized spectra of pure CH 4 , as well as mixtures of CH 4 /N 2 and CH 4 /CO 2 in molar ratios of 50/50, at pressures of 0.1, 0.5, 1, 2, 3, 4, and 5 MPa were recorded using this system. The signal-to-noise ratio in the polarized spectra of pure CH 4 was 1500 (at 0.1 MPa) and 11,000 (at 5 MPa), where the peak intensity of the ν 1 band (≈2917 cm −1 ) was the signal magnitude. The pressure measurement error was less than 1 kPa. The gas cell was thermally stabilized at 298 ± 1 K. Samples of CH 4 , N 2 , and CO 2 with a purity of greater than 99.99% were used to prepare the studied mixtures in a separate mixing chamber connected to a gas cell. Pure gases were mixed in a specified ratio of partial pressures to obtain the required molar ratio. These partial pressures were calculated from the equation of state for gases taking into account the compressibility. Compressibility factors were taken from the NIST Chemistry WebBook [46]. The molar ratio measurement error in the mixture preparation procedure is estimated within 2-3%.
The wavenumber calibration of the spectrometer was performed using the spectrum of pure CH 4 at a pressure of 0.1 MPa according to the procedure described by Brunsgaard Hansen [47]. However, the most intense lines of the ν 3 band from data of Berger [48] were taken as reference lines, instead of the emission lines of a neon lamp. As a result, the third-degree polynomial was obtained, representing the relationship between the pixel numbers of the CCD sensor and the wavenumbers of the spectrometer. The calibration error and the spectrum drift due to ambient temperature fluctuations were estimated to be less than 0.02 cm −1 . Figure 2 shows the obtained Raman spectra of pure CH 4 at various pressures in the spectral range of 2810-3030 cm −1 . The polarized spectrum is the high-intensity peak formed by closely spaced rotational-vibrational lines of the Q branch of the ν 1 band. This peak is overlapped by the O, P, and Q branches of the ν 3 band and the Q branch of the ν 2 + ν 4 band. The contribution of other overtones and hot transitions can be neglected in this range. The vibrations ν 1 and ν 2 + ν 4 are characterized by extremely weak anisotropic polarizability properties. Hereby, the ν 2 + ν 4 band is not observed in the depolarized spectra, and the ν 1 band is a low-intensity peak. An increase in medium pressure leads to the broadening of all lines due to the collisional broadening effect. Therefore, the ν 3 band is an almost continuous spectrum at a pressure of 5 MPa. However, this effect is not so pronounced for the ν 1 band, since the processes of collisional line mixing dominate here [26]. The ν 1 peak shifts to the region of low wavenumbers as the pressure increases, which corresponds to the data of [22,27,28,49,50]. The effect of the N 2 and CO 2 environments leads to different broadening and shifts of the CH 4 lines. Nevertheless, the spectrum of the mixture is similar to that of pure CH 4 at a different pressure. This difference is more pronounced as the pressure increases. As shown in Figure 3, the presence of N 2 in the mixture leads to a narrowing of the ν 1 peak, while the presence of CO 2 leads to a broadening. It is also worth noting that the N 2 environment leads to a smaller shift of the ν 1 peak to the region of low wavenumbers than CH 4 or CO 2 . These observations are in agreement with results presented in [22,27,51]. The contribution of the N 2 and CO 2 bands is negligible within the spectral range under investigation in comparison with the ν 3 and ν 2 + ν 4 lines.

Measurement Procedure
The observed depolarization ratio of an arbitrary vibrational band can be defined by Equation (1), where E ⊥ (ω) and E (ω) are the intensities of the experimental Raman spectra at the wavenumber ω, when the polarization planes of the scattered and exciting radiation are parallel (polarized spectrum) and perpendicular (depolarized spectrum), respectively. Here, it is necessary to take into account the overlap of the ν 3 and ν 2 + ν 4 bands at different pressures and environments to correctly measure the integrated intensity of the ν 1 band. The method of simulating the Raman spectrum as a sum of the profiles of each rotational-vibrational line was used for this purpose. A detailed description of this approach can be found in our previous work [33]. The positions and intensities of the ν 3 and ν 2 + ν 4 lines were taken from the study of Ba et al. [52], and the pressure broadening and shift coefficients were used the same as those in [33]. According to the features of the polarizability anisotropy, only the ν 3 lines were used to simulate the depolarized spectra. The ν 3 and ν 2 + ν 4 lines were used to simulate the polarized spectra. The influence of the N 2 and CO 2 environments on the ν 3 and ν 2 + ν 4 bands of CH 4 was imitated by simulating the spectrum at a different pressure. The integrated intensities of the depolarized and the polarized ν 1 band (E ⊥ (ν 1 ), E (ν 1 )) were measured in the range of 2880-2950 cm −1 in each experimental spectrum after subtracting the simulated spectrum (see Figure 4).

Measurement Procedure
The observed depolarization ratio of an arbitrary vibrational band can be defined by Equation (1), where ( ) are the intensities of the experimental Raman spectra at the wavenumber ω, when the polarization planes of the scattered and exciting radiation are parallel (polarized spectrum) and perpendicular (depolarized spectrum), respectively. Here, it is necessary to take into account the overlap of the ν3 and ν2 + ν4 bands at different pressures and environments to correctly measure the integrated intensity of the ν1 band. The method of simulating the Raman spectrum as a sum of the profiles of each rotationalvibrational line was used for this purpose. A detailed description of this approach can be found in our previous work [33]. The positions and intensities of the ν3 and ν2 + ν4 lines were taken from the study of Ba et al. [52], and the pressure broadening and shift coefficients were used the same as those in [33]. According to the features of the polarizability anisotropy, only the ν3 lines were used to simulate the depolarized spectra. The ν3 and ν2+ν4 lines were used to simulate the polarized spectra. The influence of the N2 and CO2 environments on the ν3 and ν2 + ν4 bands of CH4 was imitated by simulating the spectrum at a different pressure. The integrated intensities of the depolarized and the polarized ν1 were measured in the range of 2880-2950 cm −1 in each experimental spectrum after subtracting the simulated spectrum (see Figure 4).  Further, it is necessary to take into account the fluctuations of the laser power to improve the accuracy of the intensity measurement. We used the Q branch of the ν3 band of CH4 for this purpose since the ρ of this band does not depend on pressure in the range of 0-5 MPa and equals 0.75 [42,53]. Thus, the ρ(ν1) values were obtained using Equations (2) and (3): Further, it is necessary to take into account the fluctuations of the laser power to improve the accuracy of the intensity measurement. We used the Q branch of the ν 3 band of CH 4 for this purpose since the ρ of this band does not depend on pressure in the range of 0-5 MPa and equals 0.75 [42,53]. Thus, the ρ(ν 1 ) values were obtained using Equations (2) and (3): where E ⊥ (ν 3 ) and E (ν 3 ) are the integrated intensities of the Q branch of the ν 3 band in the depolarized and polarized spectra, respectively. These intensities were measured in the range of 3000-3030 cm −1 . The data obtained are presented in Figure 5. The values of the ρ(ν 1 ) are in the range of 0.0009-0.001 and the influence of the molecular environment in the pressure range of 0-5 MPa is not observed. The double standard deviation of all measurements is less than 0.0001. It should be noted that much larger values of the ρ(ν 1 ) at 5 MPa were obtained by other authors [40][41][42]. We suppose that this discrepancy is caused by the neglect or incorrect accounting of the overlap of the ν 1 peak by the ν 3 and ν 2 + ν 4 bands, in addition to the low signal-to-noise ratio. The obtained values of the ρ(ν 1 ) of pure CH 4 , where the subtraction procedure of the simulated spectra was not performed, are also shown in Figure 5 for comparison. It can be seen that the pressure dependence of the ρ(ν 1 ) is observed in this case, which corresponds to the previous results [42]. The reason for this is that the contribution of the ν 3 and ν 2 + ν 4 lines to the intensity of the ν 1 band (in the 2910-2925 cm −1 range) increases due to the collisional broadening effect. Thus, the data in Figure 5 confirm that the contribution of depolarized lines must be taken into account to obtain the most reliable values of the ρ. Further, it is necessary to take into account the fluctuations of the laser power to improve the accuracy of the intensity measurement. We used the Q branch of the ν 3 band of CH 4 for this purpose since the ρ of this band does not depend on pressure in the range of 0-5 MPa and equals 0.75 [42,53]. Thus, the ρ(ν 1 ) values were obtained using Equations (2) and (3): where E ⊥ (ν 3 ) and E (ν 3 ) are the integrated intensities of the Q branch of the ν 3 band in the depolarized and polarized spectra, respectively. These intensities were measured in the range of 3000-3030 cm −1 . The data obtained are presented in Figure 5. The values of the ρ(ν 1 ) are in the range of 0.0009-0.001 and the influence of the molecular environment in the pressure range of 0-5 MPa is not observed. The double standard deviation of all measurements is less than 0.0001. It should be noted that much larger values of the ρ(ν 1 ) at 5 MPa were obtained by other authors [40][41][42]. We suppose that this discrepancy is caused by the neglect or incorrect accounting of the overlap of the ν 1 peak by the ν 3 and ν 2 + ν 4 bands, in addition to the low signal-to-noise ratio. The obtained values of the ρ(ν 1 ) of pure CH 4 , where the subtraction procedure of the simulated spectra was not performed, are also shown in Figure 5 for comparison. It can be seen that the pressure dependence of the ρ(ν 1 ) is observed in this case, which corresponds to the previous results [42]. The reason for this is that the contribution of the ν 3 and ν 2 + ν 4 lines to the intensity of the ν 1 band (in the 2910-2925 cm −1 range) increases due to the collisional broadening effect. Thus, the data in Figure 5 confirm that the contribution of depolarized lines must be taken into account to obtain the most reliable values of the ρ.
Molecules 2022, 27, x FOR PEER REVIEW 8 of 17 caused by the neglect or incorrect accounting of the overlap of the ν1 peak by the ν3 and ν2+ν4 bands, in addition to the low signal-to-noise ratio. The obtained values of the ρ(ν1) of pure CH4, where the subtraction procedure of the simulated spectra was not performed, are also shown in Figure 5 for comparison. It can be seen that the pressure dependence of the ρ(ν1) is observed in this case, which corresponds to the previous results [42]. The reason for this is that the contribution of the ν3 and ν2 + ν4 lines to the intensity of the ν1 band (in the 2910-2925 cm −1 range) increases due to the collisional broadening effect. Thus, the data in Figure 5 confirm that the contribution of depolarized lines must be taken into account to obtain the most reliable values of the ρ. Figure 5. Depolarization ratio of the ν1 band of CH4 as a function of pressure at different molecular environments, where label a denotes the data obtained after the subtraction procedure of the simulated spectrum of the ν3 and ν2 + ν4 bands, and label b is the data obtained without the subtraction.

Uncertainty Evaluation
According to Figure 5, the measured value of the ρ(ν1) is not equal to zero even at a pressure of 0.1 MPa, which does not agree with theoretical calculations [35,53,54]. Let us estimate the error of our measurements. The main sources of the measurement error are imperfect polarization of the laser radiation, different transmittance of the polarizer in orthogonal orientations, and polarization scrambling by the windows of the gas cell [55,56], as well as the non-zero collection angle for the scattered radiation [38,[57][58][59]. The additional experiment was carried out to evaluate the influence of the first three effects. Laser radiation was directed through the cell windows and the polarizer and was guided Figure 5. Depolarization ratio of the ν 1 band of CH 4 as a function of pressure at different molecular environments, where label a denotes the data obtained after the subtraction procedure of the simulated spectrum of the ν 3 and ν 2 + ν 4 bands, and label b is the data obtained without the subtraction.

Uncertainty Evaluation
According to Figure 5, the measured value of the ρ(ν 1 ) is not equal to zero even at a pressure of 0.1 MPa, which does not agree with theoretical calculations [35,53,54]. Let us estimate the error of our measurements. The main sources of the measurement error are imperfect polarization of the laser radiation, different transmittance of the polarizer in Figure 5. Depolarization ratio of the ν 1 band of CH 4 as a function of pressure at different molecular environments, where label a denotes the data obtained after the subtraction procedure of the simulated spectrum of the ν 3 and ν 2 + ν 4 bands, and label b is the data obtained without the subtraction.

Uncertainty Evaluation
According to Figure 5, the measured value of the ρ(ν 1 ) is not equal to zero even at a pressure of 0.1 MPa, which does not agree with theoretical calculations [35,53,54]. Let us estimate the error of our measurements. The main sources of the measurement error are imperfect polarization of the laser radiation, different transmittance of the polarizer in orthogonal orientations, and polarization scrambling by the windows of the gas cell [55,56], as well as the non-zero collection angle for the scattered radiation [38,[57][58][59]. The additional experiment was carried out to evaluate the influence of the first three effects. Laser radiation was directed through the cell windows and the polarizer and was guided to the photodetector at the output (see Figure 6). At the first stage, the cell was pressurized by pure CH 4 at 0.1 MPa and the power of the transmitted radiation was measured in two orthogonal polarization orientations. At the second stage, the pressure of CH 4 in the cell was increased to 5 MPa and similar measurements were performed. It was found that the ratio P /P ⊥ was more than 1000 in both cases, where P ⊥ and P are the measured radiation power with perpendicular and parallel orientation of the polarization plane to the polarization plane of the exciting radiation, respectively. It should be noted that the entrance window (W 1 ) and the exit window (W 2 ) influenced the results obtained in this experiment, but the window W 1 and the side window (W 3 ) influenced the measurements of the ρ(ν 1 ). Since all the cell windows are identical, we can conclude that the systematic measurement error of the ρ(ν 1 ) is less than 0.001 at a zerocollection angle, taking into account the aforementioned effects of polarization scrambling.  The approach based on the calculations presented by Schlösser et al. [57] was used to estimate the measurement error in the case of the non-zero collection angle. A detailed description of the calculations performed is provided in Appendix A of this study. As a result, the geometric effect introduces the systematic measurement error of no more than 2% of the ρ(ν1) = 0.001, without taking into account the effects of polarization scrambling. Since the nonzero angle effect has a small contribution, the estimate of the total systematic measurement error of the ρ(ν1) is less than 0.001. Therefore, we can conclude that the true depolarization ratio of the ν1 band of CH4 is within 0-0.001 in the pressure range of 0.1-5 MPa.

Conclusions
In this study, the depolarization ratio of the ν1 band of CH4 was measured using the Raman spectrometer that combines both high resolution and high sensitivity. It was found that the depolarization ratio of the ν1 peak of pure CH4 or perturbed by the N2/CO2 molecular environment did not exceed 0.001 in the pressure range of 0.1-5 MPa. This value is significantly less than the measurements reported in earlier studies. In our view, this discrepancy is a consequence of correctly taking into account the overlap of the ν1 band by the ν3 and ν2 + ν4 bands using the spectra simulation in this study. These results imply that the correction of the tensor components of the total polarizability derivative of CH4 due to the effect of the N2/CO2 environment, and pressure can be neglected in the pressure range of 0.1-5 MPa. Therefore, the line intensities of CH4 in vacuum calculated using the tensor formalism approach are suitable for simulating its spectra in the field of Raman gas analysis of methane-bearing media (e.g., fuel gases, atmospheric air, exhaled air, etc.).  The approach based on the calculations presented by Schlösser et al. [57] was used to estimate the measurement error in the case of the non-zero collection angle. A detailed description of the calculations performed is provided in Appendix A of this study. As a result, the geometric effect introduces the systematic measurement error of no more than 2% of the ρ(ν 1 ) = 0.001, without taking into account the effects of polarization scrambling. Since the non-zero angle effect has a small contribution, the estimate of the total systematic measurement error of the ρ(ν 1 ) is less than 0.001. Therefore, we can conclude that the true depolarization ratio of the ν 1 band of CH 4 is within 0-0.001 in the pressure range of 0.1-5 MPa.

Conclusions
In this study, the depolarization ratio of the ν 1 band of CH 4 was measured using the Raman spectrometer that combines both high resolution and high sensitivity. It was found that the depolarization ratio of the ν 1 peak of pure CH 4 or perturbed by the N 2 /CO 2 molecular environment did not exceed 0.001 in the pressure range of 0.1-5 MPa. This value is significantly less than the measurements reported in earlier studies. In our view, this discrepancy is a consequence of correctly taking into account the overlap of the ν 1 band by the ν 3 and ν 2 + ν 4 bands using the spectra simulation in this study. These results imply that the correction of the tensor components of the total polarizability derivative of CH 4 due to the effect of the N 2 /CO 2 environment, and pressure can be neglected in the pressure range of 0.1-5 MPa. Therefore, the line intensities of CH 4 in vacuum calculated using the tensor formalism approach are suitable for simulating its spectra in the field of Raman gas analysis of methane-bearing media (e.g., fuel gases, atmospheric air, exhaled air, etc.).
Author Contributions: Supervision, project administration, funding acquisition, validation, and writing-review and editing, D.V.P.; formal analysis, data curation, visualization, and writingoriginal draft, A.S.T.; conceptualization, investigation, and methodology, D.V.P. and A.S.T. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.
Sample Availability: Not applicable.

Appendix A
The integrated intensity of an arbitrary vibrational Raman band v j→k observed in the direction (ϕ,θ) can be expressed as follows [60]: where E in is the intensity of the incident radiation exciting the Raman scattering; Φ j is the scattering strength function of the scattered radiation in the direction (ϕ,θ); ϕ and θ are the scattering angles; N j and g j are the population and the degeneracy of the jth vibrational energy level; and I p and S p are the polarization states of the plane-polarized incident and scattered radiation (p = ⊥ or ), respectively (see Figure A1 for details). The combinations of the polarization states assign the four scattering strength functions: Φ(S ⊥ , I ⊥ , ϕ, θ) = 45α 2 cos 2 ϕ + γ 2 3 + cos 2 ϕ , (A2) Φ(S , I ⊥ , ϕ, θ) = 45α 2 cos 2 θ sin 2 ϕ + γ 2 3 + cos 2 θ sin 2 ϕ , (A3) Φ(S ⊥ , I , ϕ, θ) = 45α 2 sin 2 ϕ + γ 2 3 + sin 2 ϕ , (A4) Φ(S , I , ϕ, θ) = 45α 2 cos 2 θ cos 2 ϕ + γ 2 3 + cos 2 θ cos 2 ϕ , where α and γ are the mean and the anisotropy of the total polarizability derivative, respectively, with respect to the normal coordinate of the molecule related to the vibration v j→k . For example, Φ(S ⊥ , I , ϕ, θ) is the scattering strength function of the perpendicular plane-polarized scattered radiation observed in the direction (ϕ,θ), where the Raman scattering is induced by parallel plane-polarized radiation. The corrected depolarization ratio (ρ corr ) of the band, independent of the collection angle and in the case of perfectly plane-polarized radiation, is defined as the ratio of intensities of the perpendicular and parallel polarized scattered radiation observed from a single point at a zero solid angle in the direction (ϕ = 0, θ = π/2). Taking into account the introduced designations, the corrected depolarization ratio is given as: ( , , , ) 45 cos cos 3 cos cos where α and γ are the mean and the anisotropy of the total polarizability derivative, respectively, with respect to the normal coordinate of the molecule related to the vibration vj→k. For example, ( , , , ) is the scattering strength function of the perpendicular plane-polarized scattered radiation observed in the direction (φ,θ), where the Raman scattering is induced by parallel plane-polarized radiation. According to theoretical calculations reported by Abbate et al. [35], the anisotropy γ of the ν 1 band of CH 4 is zero, and hence, ρ corr (ν 1 ) = 0. Besides, α = 0 for the ν 3 band and ρ corr (ν 3 ) = 0.75, respectively. Let us suppose that the scattered radiation is collected over a finite solid angle Ω and from a region of active molecules characterized by a finite volume V. Then, Equation (A1) is given as: Let us assume that the exciting radiation in the V region is not perfectly plane-polarized due to the introduced polarization distortions and the imperfect polarization of the radiation source. Then, the exciting radiation can be considered as a superposition of two waves with mutually orthogonal polarization: where E ⊥ in and E in are the intensities of the perpendicular and parallel plane-polarized exciting radiation, respectively. Therefore, Raman scattering can be described as a superposition of two luminous fluxes, each of which is excited by radiation E ⊥ in and E in . In this case, the observed depolarization ratio (ρ obs ) is expressed as follows: where η 1 is the ratio E ⊥ in /E in . If the collected radiation from the region V undergoes polarization distortions introduced by the collection system, the scattered intensity can also be represented as a superposition on the analogy of Equation (A8). Let us assume that the initially parallel polarized scattered radiation is split into two waves with mutually orthogonal polarization as: where E obs and E ⊥ obs are the intensities of the parallel and the perpendicular planepolarized observed radiation from the initially parallel polarized scattered radiation, respectively. Then, the separated wave E ⊥ obs contributes to the intensity of the perpendicular polarized observed radiation at the end of the collection system. Conversely, the separated wave E obs⊥ from the initially perpendicular polarized scattered radiation contributes to the intensity of the parallel polarized observed radiation. Therefore, Equation (A9) takes the following form: If splitting into two waves occurs in the same ratio for the E obs and E ⊥ obs , that is: then Equation (A11) is expressed as follows: where: The Raman spectrometer used in this work detects only those scattered rays that propagate through the collecting lens and are focused into the region of the entrance slit (4 mm × 30 µm) of the spectrometer (see Figures 1 and A1). Lenses used to collect the scattered radiation magnify the image of the object twice. Hence, the effective scattering area is about 2 mm × 15 µm (cross-section of the region V in the focal plane of the collecting lens). Therefore, the intensity distribution of the laser beam cross-section and the width (15 µm) of the effective area can be neglected. Hereby, the integration over the solid angle Ω and the region V can be replaced by the integration over ϕ, θ, and z in Equations (A14) and (A15). Substituting Equation (A6) in Equations (A2)-(A5), the simplified Equation (A13) as a function of the corrected ρ can be derived: A = η 1 (CS + ρ corr (U − CS)) + (CC + ρ corr (U − CC)), (A16) B = η 1 (C + ρ corr (U − C)) + (S + ρ corr (U − S)), These expressions are obtained with the proviso that the aperture of the focusing lens does not limit the collection of the scattered radiation from the region of z = 0 (D 2 > D 1 ). Moreover, the aperture and position of the exit window of the gas cell are neglected for the same reason.
As discussed in Section 3.3., the polarization scrambling effects of scattered radiation propagating through the path of W 1 → W 3 → Polarizer or W 1 → W 2 → Polarizer are similar (see Figure 6). Thus, the approximate equality η 1 ≈ η 2 ≥ 1000 holds in Equations (A13), (A16) and (A17). In other words, the perpendicular plane-polarized laser radiation will contain 1 part of the parallel polarized light to more than 1000 perpendicular polarized parts, at the end of the collection system. Figure A2 shows the deviation of the ρ obs from the ρ corr as a function of ρ corr , parameter η, and the full collection angle (2ϕ max ). These values are calculated using Equations (A13) and (A16)-(A25), and the parameters from Table 1. As expected, the collection of radiation using the lens with a larger f-number leads to a greater deviation of the ρ. However, the deviation of the ρ tends to a value of 1/η in the vicinity of the zero-collection angle. This deviation is caused by the splitting of the plane-polarized scattered radiation as 1 to η due to the polarization scrambling effect, even in the case of collecting from a single point at a zero solid angle. In turn, the attenuation of the polarization scrambling effect leads to less deviation of the ρ at a fixed collection angle. The deviation of the ρ is approximately 0.001 at the η = 1000 and 2ϕ max = 14.25 • .